In my May 2004 article, "Fellowship and Admission Decisions for Graduate Programs in the Mathematical Sciences," I addressed exogenous factors associated with the process of acceptance or rejection by graduate school programs in mathematics. Because being rejected is not the most thrilling experience, I highlighted some little-known nonacademic factors that can play a key role in the process. One's energy is better spent selecting the best available program, because often it's hard to know the reasons behind a rejection letter. In the end, it is better to move passionately forward with the options that one has.
Once a student has been accepted by a program, the next natural question that arises is how he or she should proceed during the critical first year of graduate studies. It would be impossible to offer up a plan that works for all areas, or even for a single field of expertise. Effective plans are not necessarily university- and program-specific. The comments that follow should not be taken unquestioningly; at best, they provide a model that may be useful in the design of a student's initial plan of action.
The model, or philosophy, that I will describe derives from my experiences at private and state universities, as a graduate student, postdoctoral fellow, and faculty member. My expertise is in the fields of mathematics (applied and theoretical mathematics and computational biology), but the overall themes apply to all graduate students, regardless of discipline.
Initial Plan of Attack
In biology, students often have a clear idea of what they want to do; they have a long-term research plan. This is not often the case in mathematics or applied mathematics. Often these students put too much emphasis on differentiating between "applied" and "pure" mathematics, and in the process they may lock themselves into a program they may find to be too restrictive or in some other way not fully satisfying.
Most students who earn an undergraduate degree in mathematics at a U.S. liberal arts college assume that pure mathematics is something abstract, like algebra, and that the term "applied mathematics" describes something mundane, like numerical analysis. The fact that both fields are closely connected is not yet part of the students' vision. Hence, avoiding either pure or applied mathematics courses may not be the best strategy at the beginning of graduate school. It is usually better to take courses in both.
In the past, most students entering a graduate mathematics program had to become highly proficient in three areas: algebra, topology, and analysis. Certainly, a solid training in these areas will make you a solid mathematician, but focusing just on these areas may not help you discover your true calling. Students who are interested in applications, for example, should consider taking sequences in probability or statistics or both, differential equations/dynamical systems, numerical analyses/computational linear algebra/optimization, combinatorics/graph theory, logic/computer science, and applied mathematics/mathematical biology--to name just a few. Students will learn useful and challenging mathematics through these sequences.
Students should not forget that many people have strong opinions, and that mathematicians are no exception. Of course, if you receive or ask for advice from graduate students or faculty members, take it seriously. But also be sure to base your decisions on what you think is best for you--not on something that's on somebody's recommended "typical" plan. Graduate school is for those who have a passion for learning. This passion, and your own curiosity, should play a major role in your decisions.
Keep in mind too that the paths taken by successful individuals are not likely the paths they would choose today. An "optimal" path for me today, for instance, would include a different set of classes than the ones I actually took. I never took courses in dynamical systems, ordinary differential equations, or mathematical biology--the very areas that have been at the heart of my research for the past 18 years! And I am not a special case; plenty of mathematicians have shifted gears.
Selecting an Adviser
Selecting a supportive adviser for your Ph.D. is the most important academic and professional decision that you will make in graduate school. If you selected a university because of its strength in a particular area of mathematics, then a pool of potential Ph.D. advisers should be available to you there. You should talk to most, if not all, of the faculty in your field of interest; take courses with as many of them as you can during your first 2 years; and attend the field seminar and read some of the faculty's publications before making a decision. At the same time, don't forget that this is a mutual decision; a faculty member may not want to take you as his or her student. If this proves to be the case, there could be many reasons behind the decision. Better just to find another adviser.
Of course, it's always possible to change advisers, but it's better to make the right choice the first time around. If you don't have to choose right away, don't rush. On the other hand, don't postpone selecting an adviser for more than 2 years. Ideally--although this is not always possible--you should have an adviser in place by the end of your first year of graduate studies.
If you've chosen your school because of a particular researcher, be sure to visit the institution before you accept its offer (particularly if you have just one adviser in mind) to make sure that individual is aware you are coming. During your visit, talk as well to your prospective mentor's students, so that you'll understand upcoming expectations. Advisers expect committed students--students who place their graduate work as their main priority. Advisers know that you need encouragement and support, but they also expect passion and commitment.
There are no standard rules for working with advisers. Some advisers demand weekly meetings, whereas others will meet you only when necessary. Some advisers will demand tight control over your research, whereas others will expect you to take the lead. Being sure that your adviser's style will work for you is essential.
In part 2 of this piece, I will comment further about the first year of graduate school, touching on matters such as social activities and my own experiences as a graduate student.
Carlos Castillo-Chavez is a Joaquin Bustoz Jr. Professor of Mathematical Biology at Arizona State University. He can be reached at email@example.com.